Theorem 19.29x 1804

Description: Variation of 19.29 1801 with mixed quantification. (Contributed by NM, 11-Feb-2005.)

Assertion

Ref	Expression
19.29x	⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Proof of Theorem 19.29x

Step	Hyp	Ref	Expression
1		19.29r 1802	. 2 ⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓))
2		19.29 1801	. . 3 ⊢ ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑 ∧ 𝜓))
3	2	eximi 1762	. 2 ⊢ (∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
4	1, 3	syl 17	1 ⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)